Rational Noncrossing Partitions for all Coprime Pairs

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by  Michelle Bodnar

RATIONAL NONCROSSING PARTITIONS FOR ALL COPRIME PAIRS
MICHELLE BODNAR
Abstract. Forcoprimepositiveintegersa<b,Armstrong,Rhoades,andWilliams(2013)defined
7 asetNC(a,b)ofrationalnoncrossingpartitions,asubsetoftheordinarynoncrossingpartitionsof
1 {1,...,b−1}. BodnarandRhoades(2015)confirmedtheirconjecturethatNC(a,b)isclosedunder
0 rotationandprovedaninstanceofthecyclicsievingphenomenonforthisrotationaction. Wegive
2 a definition of NC(a,b) which works for all coprime a and b and prove closure under rotation and
cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all
b
e coprime a and b, and provide a character formula for the action of Sa×Zb−1 on ParkNC(a,b).
F

6 1. Introduction

] Let W be a Weyl group with root lattice Q, degrees d ,d ,...,d , and Coxeter number h = d .
O 1 2 (cid:96) (cid:96)
Then W acts on the “finite torus” Q/(h+1)Q. Cosets in Q/(h+1)Q give a model for parking
C
functions attached to W [3]. It has been shown by Haiman [6] that the number of orbits of this
.
h action is given by
t
a (cid:89) h+di
Cat(W) := ,
m
d
i
i
[
whichhascometobeknownastheCoxeter-CatalannumberofW. Moregenerally, ifpisapositive

2
integer which is coprime to the Coxeter number h, Haiman [6] showed that the number of orbits in
v
8 the action of W on Q/pQ is
9 (cid:89) p+di−1
1 Cat(W,p) = .
d
7 i
i
0
This number has come to be known as the rational Catalan number of W at parameter p.
.
1
0
On the level of Weyl groups the Catalan and Fuss-Catalan objects, obtained by taking p = h+1
7
1 and mh+1 respectively, have been defined and studied [1]. When W = Sa is the symmetric group,
: we have
v
(cid:18) (cid:19)
i 1 2a
X Cat(Sa,a+1) = = Cat(a)
a+1 a
r
a where Cat(a) is the classical Catalan number, famously counting noncrossing partitions, Dyck
paths, well-paired parentheses, as well as hundreds of other combinatorial objects. Furthermore,
we have
(cid:18) (cid:19)
1 ka+a+1
Cat(S ,ka+1) = = Cat(k)(a)
a
ka+a+1 a
where Cat(k)(a) is the Fuss-Catalan number, counting generalizations of Catalan objects such as
noncrossing partitions whose block sizes are all divisible by k. However, it wasn’t until 2013 that
Armstrong et. al [4, 2] undertook a systematic study of type A rational Catalan combinatorics.
For coprime positive integers a and b, the rational Catalan number is
(cid:18) (cid:19)
1 a+b
Cat(S ,b) = = Cat(a,b).
a
a+b a,b
1
2 MICHELLEBODNAR
Observe that Cat(n,n+1) = Cat(n), so that rational Catalan numbers are indeed a generalization
of the classical Catalan numbers. The program of rational Catalan combinatorics seeks to gener-
alize Catalan objects such as Dyck paths, the associahedron, noncrossing perfect matchings, and
noncrossing partitions (each counted by the classical Catalan numbers) to the rational setting. For
instance, Cat(a,b) counts the number of a,b-Dyck paths, NE-lattice paths from the origin to (b,a)
staying above the line y = ax.
b
For coprime parameters a < b, Armstrong et. al [4] defined the a,b-noncrossing partitions,
NC(a,b), to be a subset of the collection of noncrossing partitions of [b−1] arising from a laser
construction involving rational Dyck paths. A characterization of these rational noncrossing parti-
tions was given in [5], where it was shown that NC(a,b) is closed under dihedral symmetries and
thattheactionofrotationonNC(a,b)exhibitsacyclicsievingphenomenon. Additionally, amodel
for a,b-noncrossing parking functions was given which carries an S ×Z action, and a character
a b−1
formula was stated and proved. However, this rational generalization and others relies on the fact
that a < b.
It is of intrinsic combinatorial interest to see whether such results hold in the case where a > b.
Moreover, this seems reasonable as Haiman’s formula holds for any coprime pair a,b. Furthermore,
we are motivated by the favorable representation theoretic properties of the rational Cherednik
algebra attached to the symmetric group S at parameter b/a. Such properties persist even when
a
a > b. It is thus desirable to remove the condition a < b and define rational noncrossing partitions
for all coprime pairs (a,b). This paper provides the first type A combinatorial model for rational
Catalan objects defined for all coprime a and b, along with proofs of analogous results to those
given in [5].
The rest of the paper is organized as follows: Section 2 begins with background information on
rationalDyckpathsandnoncrossingpartitions. InSection 3wegiveaninterpretationofa,b-Dyck
paths in terms of pairs of labeled noncrossing partitions which generalizes the laser construction
introduced in [4] and prove that such pairs of partitions are closed under a suitable rotation action.
In Section 4 we provide a generalization of rank from [5] and show that block rank commutes with
rotation. Wealsoprovideacharacterizationofwhenagivenlabeledpairofnoncrossingpartitionsis
an element of NC(a,b). Section 5 introduces d-modified rank sequences and goes through a series
of lemmas which ultimately allow us to count the number of elements in NC(a,b) which are invari-
ant under d-fold rotation. In Section 6, we prove various refinements of cyclic sieving results for
NC(a,b). A step in this direction has already been made by Thiel in [13] where cyclic sieving was
shown in the case (a,b) = (n+1,n) by considering objects called noncrossing (1,2)-configurations.
Wewillrevisitthisconstructionandgiveabijectionbetweenourn+1,n-noncrossingpartitionsand
these (1,2) configurations. Finally, in Section 7, we generalize a,b-noncrossing parking functions,
ParkNC(a,b), to all coprime a and b and prove a character formula for the action of S ×Z on
a b−1
ParkNC(a,b).
2. Background
2.1. Rational Dyck Paths. Let a and b be coprime positive integers. An a,b-Dyck path D is a
lattice path in Z2 consisting of unit length north and east steps which starts at (0,0), ends at (b,a),
and stays above the line y = ax. By coprimality, the path will never touch this line. For example,
b
consider the 7,4-Dyck path NNNENENNENE is shown in Figure 1. The a,b-Dyck paths are
counted by the rational Catalan number Cat(a,b) = 1 (cid:0)a+b(cid:1). A vertical run of D is a maximal
a+b a
contiguous sequence of north steps. The 7,4-Dyck path shown in Figure 1 has 4 vertical runs of
RATIONAL NONCROSSING PARTITIONS FOR ALL COPRIME PAIRS 3
Figure 1. 7,4-Dyck Path NNNENENNENE with the line y = 7x
4
lengths3, 1, 2, and1respectively. Avalley ofD isalatticepointponD suchthatpisimmediately
preceded by an east step and succeeded by a north step. Figure 1 has three valley points. When
(a,b) = (n,n+1), rationalDyckpathsareequivalenttoclassicalDyckpaths, NE-latticepathsfrom
(0,0) to (n,n) which stay weakly above the line y = x, and are counted by the classical Catalan
numbers.
2.2. Noncrossing Partitions. A set partition π of [n] := {1,2,...,n} is noncrossing if its blocks
do not cross when drawn on a disk whose boundary is labeled clockwise with the number 1, 2, ...,
n. Equivalently, π is noncrossing if there do not exist a < b < c < d such that a and c are in the
same block B, and b and d are in the same block B(cid:48) (cid:54)= B. Let NC(n) denote the set of noncrossing
partitions of [n]. Such partitions are counted by the classical Catalan numbers
(cid:18) (cid:19) (cid:18) (cid:19)
1 2n 1 2n+1
Cat(n) = = = |NC(n)|.
n+1 n 2n+1 n
(cid:18) (cid:19)
1 2 ··· n−1 n
The rotation operator rot acts on the set NC(n) by the permutation .
2 3 ··· n 1
A labeled noncrossing partition is a noncrossing partition with a nonnegative integer called a la-
bel attached to each block. When we apply rot to a labeled noncrossing partition the elements
ofeachblockshiftasintheunlabeledcase, andblockmaintaintheirlabelsthroughouttherotation.
3. Construction and Properties of NC(a,b)
3.1. Rational Pairs of Noncrossing Partitions. A simple bijection maps classical Dyck paths
to noncrossing partitions. The same map, when a < b, sends a,b-Dyck paths to a,b-noncrossing
partitions [5]. We’ll now define a more general version of this map, π, that makes sense for any
a,b-Dyck and use this map to define rational a,b-noncrossing partitions for any coprime a and b.
Let D be an a,b-Dyck path and label the east ends of the nonterminal east steps of D from left to
right with the numbers 1,2,...,b−1. Let p be the label of a lattice point at the bottom of a north
step of D. The laser (cid:96)(p) is the line segment of slope a which fires northeast from p and stops the
b
next time it intersects D. By coprimality, (cid:96)(p) terminates on the interior of an east step of D. For
instance, consider the 10,7-Dyck path shown on the left in Figure 2. We have that (cid:96)(3) hits D on
the interior of the east step whose right endpoint is labeled 6. We define the laser set (cid:96)(D) to be
the set of pairs (i,j) such that D contains a laser starting at label i and which terminates on an
4 MICHELLEBODNAR
east step with west x-coordinate j. For the Dyck path in Figure 2 we have
(cid:96)(D) = {(2,6),(3,5),(5,5),(6,6)}.
Define a pair of labeled noncrossing partitions π(D) = (P,Q) as follows: fire lasers from all
labeled points which are also at the bottom of a north step. As is done in [5], we define the
partition P by the visibility relation
i ∼ j if and only if the labels i and j are not separated by laser fire.
P
Label each block of P by the length of the vertical run immediately preceding the minimal element
of the block. We will refer to this as the rank of the block. Call a vertical run a P-rise if it has
length greater than a. Other than the block labeling, this map is the same as the one described in
b
[5]. We will now describe the creation of the blocks of Q, a genuinely new feature of this map.
We call a vertical run a Q-rise if it has nonzero length which is less than a. In the special case
b
where a < b, there can be no Q-rises since a/b < 1. In Figure 2, the first three vertical runs are
P-rises and the last two are Q-rises. For a given Q-rise, let E denote the east step immediately
following it. We define the partition Q by the relation
i ∼ j if and only if (cid:96)(i) and (cid:96)(j) hit the same east step immediately following a Q-rise.
Q
We label the blocks of Q as follows: If B is a block of Q and i ∈ B, then we label B with the
number of north steps beneath the west endpoint of the east step hit by (cid:96)(i). As with P, we will
call this block labeling the rank of the block. This labeling is well-defined since the lasers fired
from every element of B necessarily hit the same east step. It should also be noted that there will
often be singleton blocks of rank 0. We will call these singletons of rank 0 the trivial blocks of Q,
and often omit them when describing Q. We will refer to blocks of Q whose ranks are positive as
nontrivial blocks. Let π(D) denote the labeled pair (P,Q) associated to D under this construction.
Figure 2. A 10,7-Dyck path with corresponding pair of labeled noncrossing partitions
Figure 2 shows a 10,7-Dyck path with labels and lasers drawn in. The pair (P,Q) which results,
also shown in Figure 2, is as follows: P = {{1,2},{3,6},{4,5}} with block ranks 3, 2, and 3 respec-
tively and Q = {{1},{2,6},{3,5},{4}} with block ranks 0, 1, 1, and 0 respectively. In particular,
the blocks {1} and {4} are trivial blocks of Q. The ranks are written in smaller font near the lines
indicating the block structure. In general, we will omit the trivial blocks of Q and simply write
Q = {{2,6},{3,5}}, each with rank 1. We see that Q-rises occur above the points labeled 5 and 6,
RATIONAL NONCROSSING PARTITIONS FOR ALL COPRIME PAIRS 5
which create the two nontrivial Q blocks.
Each north step contributes to the rank of either a P block or a Q block, but not both. In
particular, the length of a P-rise is the rank of a block of P, and the length of a Q-rise is the rank
of a block of Q. This implies that the sum of the ranks of the P and Q blocks is a. Note that
elements in the same block of Q are necessarily in different blocks of P, since elements in the same
block of Q are separated by the lasers which they fire to hit the common east step.
When a < b, Q contains only singletons of rank 0 and P is the rational noncrossing partition
associated to D as described in [5]. The ranks of blocks are uniquely determined by the structure
of P, which is why labeling blocks by rank was not considered in [5]. When a > b, the ranks of
a blocks are no longer uniquely determined by the structure of P and Q. For instance, the 5,3-
Dyck paths NNNENNEE and NNENNNEE both give rise to P = {{1},{2}} and only trivial
Q blocks. Thus, the rank labels are a necessary feature of the construction of π(D) which makes
themapπinjective. Wearenowreadytoprovesomeusefulpropertiesofa,b-noncrossingpartitions.
Proposition 3.1. Let (P,Q) ∈ NC(a,b). There cannot exist 1 ≤ i < b − 1 such that i is the
maximal element of a block of Q and i+1 is the minimal element of a block of P.
Proof. Let D be such that π(D) = (P,Q). If i is the maximal element of a block of Q then the
lattice point labeled i is at the bottom of a Q-rise, so its length is less than a/b. On the other,
hand if i+1 is also the minimal element of a block of P then the lattice point labeled i is at the
bottom of a P-rise, whose length must be greater than a/b, a contradiction. (cid:3)
Proposition 3.2. Let D be an a,b Dyck path and (P,Q) = π(D). Given a label p, let E denote
p
the east step whose interior is hit by (cid:96)(p). If k < k are in the same block of Q then E = E
1 2 k1 k2
and (cid:96)(k ) hits E west of (cid:96)(k ).
2 k2 1
Proof. This is clear from the definition of blocks of Q and the fact that lasers are noncrossing. (cid:3)
Proposition 3.3. Given a Dyck path D, if π(D) = (P,Q) then Q is a noncrossing partition.
Proof. This follows immediately from the definition of Q blocks. Since lasers in D do not cross and
labels increase from left to right. it is impossible to have crossing Q blocks. (cid:3)
We say that two noncrossing partitions P and P of {1,2,...,n} are mutually noncrossing if
1 2
there do not exist a < b < c < d such that a and c are in the same block of P and b and d are in
i
the same block of P for i,j ∈ {1,2} and i (cid:54)= j. Equivalently, draw the numbers 1 through n on
j
the boundary of a disk. Then P and P are mutually noncrossing if when we draw the boundary
1 2
of the convex hulls of the blocks of P with solid lines and the convex hulls of the blocks of P in
1 2
dashed lines, no solid line crosses the interior of a dashed line. Note that solid-dashed intersections
at vertices are permissible. For example, the picture on the left of Figure 3 contains P, from Figure
2, drawn with dashed lines, superimposed onto Q, drawn with solid lines. We see that there are
intersectionsonlyatlabels. Ontheotherhand, thepictureontherightinFigure3showsthatifwe
superimpose a rotated version of P onto Q, then the partitions are no longer mutually noncrossing.
In particular, the {1,4} block of P crosses both the {2,6} and {3,5} blocks of Q.
Proposition 3.4. Given a Dyck path D, if π(D) = (P,Q) then P and Q are mutually noncrossing.
Proof. Suppose that there exist a < b < c < d such that a and c are in the same block of P and b
and d are in the same block of Q. Let R denote the region bounded by D and (cid:96)(b) and S denote the
remaining of the interior of the Dyck path, bounded by (cid:96)(b) and the main diagonal. Since a and c
are in the same block of P the they must either both lie in R or both lie in S. Since a < b we must
6 MICHELLEBODNAR
Figure 3. The pair on the left is mutually noncrossing. The pair on the right is not.
have a ∈ S, which implies c ∈ S. By Proposition 3.2, (cid:96)(d) hits D west of (cid:96)(b), so (cid:96)(d) ∈ R. How-
ever, d > c and c ∈ S, so we must also have d ∈ S, a contradiction. See Figure 4 for a sketch of this.
Figure 4.
On the other hand, suppose a < b < c < d such that a and c are in the same block of Q and b
and d are in the same block of P. Let R denote the region bounded by D and (cid:96)(a) and S denote
the remaining region of the interior of D. Since b appears left of c and (cid:96)(c) is left of (cid:96)(a), we must
have b ∈ R, so d ∈ R. Now, d cannot appear to the left of where (cid:96)(c) hits D because then (cid:96)(c)
would disconnect b from d, but we know they’re in the same block. On the other hand, if d appears
to the right of c then it exists between (cid:96)(c) and (cid:96)(a), which is also impossible since (cid:96)(c) and (cid:96)(a)
hit the same east step. Therefore there can be no crossings, and we conclude that P and Q are
mutually noncrossing. (cid:3)
It now makes sense to define the set NC(a,b) of (a,b) noncrossing partitions by
NC(a,b) = {π(D)|D is an a,b-Dyck path}.
Next, we will define a rotation operator rot(cid:48) on a,b-Dyck paths that commutes with π. On other
words, such that if π(D) = (P,Q), then π(rot(cid:48)(D)) = rot−1(π(D)) where rot is the map acting
componentwise on P and Q sending i to i+1, modulo b−1, which preserves ranks.
Definition 3.5. Let D = Ni1Ej1···NimEjm be the decomposition of D into nonempty vertical
and horizontal runs. We define the rotation operator rot(cid:48) as follows:
(1) If m = 1, so that D = NaEb, we set
rot(cid:48)(D) = NaEb = D.
RATIONAL NONCROSSING PARTITIONS FOR ALL COPRIME PAIRS 7
(2) If m,j > 1, we set
1
rot(cid:48)(D) = Ni1Ej1−1Ni2Ej2···NimEjm+1.
(3) If m > 1 and j = 1, let P = (1,i ) be the westernmost valley of D. The laser (cid:96)(P) fired
1 1
from P hits D on a horizontal run Ejk for some 2 < k < m. Suppose that (cid:96)(P) hits the
horizontal run Ejk on step r, where 1 ≤ r ≤ ik. There are two cases to consider:
If r = 1, we set
rot(cid:48)(D) = Ni2Ej2···Nik−1Ejk−1Ni1EjkNik+1Ejk+1···NimEjmNikE.
If r > 1, we set
rot(cid:48)(D) = Ni2Ej2···Nik−1Ejk−1NikEr−1Ni1Ejk−r+1Nik+1Ejk+1···NimEjm+1.
This definition is consistent with, but more general than, the one given in Section 3.1 [5]. The
r = 1 case in (3) will never occur if a < b but can if a > b, so this new definition is necessary. The
next proposition shows that rot(cid:48) is the path analog of rot−1 on set partitions.
Proposition 3.6. The operator rot(cid:48) defined above gives a well-defined operator on the set of a,b-
Dyck paths. Furthermore, for any Dyck path D, if π(D) = (P,Q), then π(rot(cid:48)(D)) = rot−1(π(D)).
Proof. First we must check that for any a,b-Dyck path D, rot(D) does in fact stay above the line
y = ax. The definition of this rotation operator differs from the one given in Section 3.1 of [5] for
b
a < b only in the first case in (3), so that is the only case we need to consider here. It is easiest to
explain what happens visually. In Figure 5 we break the generic Dyck path at the diagonal slashes
into 5 pieces. The segment labeled 1 is the initial vertical run. Segment 2 is the single east step
which follows. We have that (cid:96)(1) hits the first step of segment 5, which continues to the end of the
path. Segment 4 denotes the vertical run immediately preceding the east step hit by (cid:96)(1). Segment
3 denotes the segment between segments 2 and 4. The labeled path on the right shows how the
inverse rotation operator shifts these segments.
Since segment 3 stays above a laser fired in D, segment 3 in rot(cid:48)(D) must stay above the line
y = ax. Since the segment 4 is a Q-rise in D, we know that the segment 4 of rot(cid:48)(D) has length
b
at most (cid:98)a/b(cid:99), so the segments 4 and 2 of rot(cid:48)(D) stay above the line y = ax. Since segment 5
b
stays above the line in D, it is clear that it stays above the line in rot(cid:48)(D) as well. Finally, since
segment 1 is a single vertical run, it cannot cross the line. Thus, the path rot(cid:48)(D) stays above the
line y = ax so it is a valid Dyck path. Next we need to argue that π(rot(cid:48)(D)) = rot−1(π(D)). To
b
Figure 5. The Dyck path on the right is the rotated version of the path on the left
do this, we simply consider how the lasers change from D to rot(cid:48)(D).
8 MICHELLEBODNAR
(1) The lasers fired from points in segment 5 D are identical to the lasers fired in segment 5 of
rot(cid:48)(D), shifted one unit west.
(2) The lasers fired within segment 3 which hit just west of a label k in D hit just left of the
label k−1 in rot(cid:48)(D).
(3) Thelaserfromthepointlabeled1inD isreplacedbythelaserfiredfromtheendofsegment
3 in rot(cid:48)(D), so the rotated block includes b−1 instead in the rotation as desired.
(4) Letk bethelabelatthebaseofsegment4inD. Thenk and1areinthesameblockofQin
π(D). In rot(cid:48)(D), this laser is fired from k−1, and as described in (3) it hits the terminal
east step. Since segment 4 is translated to be the vertical run immediately preceding the
terminal east step, the laser fired from b−1 in rot(cid:48)(D) also hits the terminal east step, so
k −1 and b−1 are in the same block of Q in π(rot(cid:48)(D)), completing the proof that the
blocks of π(rot(cid:48)(D)) rotate as desired.
(cid:3)
It now makes sense to define rot(D) = rot(cid:48)−1(D). In other words, rot(D) is such that
π(rot(D)) = (rot(P),rot(Q)).
Given an a,b-Dyck path D, one can obtain a b,a-Dyck path τ(D) by applying the transposition
operator τ which reflects a path about the line y = −x, then shifts it such that its southern-
most point is at the origin. One might hope that transposition would commute with rotation in
the sense that τ(rot(D)) = rot(τ(D)); however, this is not the case, which can be seen immedi-
ately from an example. Let D = NNNNENENNE. If we first transpose, we obtain the path
NEENENEEEE which corresponds to the partition A = {{1,2},{3,6},{4,5}}. However, if we
first rotate D, then transpose, we obtain the partition B = {{1,6},{2,3},{4,5}}, which is not
obtainable from A via any rotation. Since the relevant information of a noncrossing partition is
read off from the vertical runs of its associated Dyck path rather than the horizontal runs, which
are not preserved under rotation, this is not surprising. However, it is more subtle to see that
NC(a,b) fails to be closed under reflection.
Itwasshownin[5]thatNC(a,b)isclosedunderthereflection operator,givenbythepermutation
(cid:18) (cid:19)
1 2 ··· b−2 b−1
rfn =
b−1 b−2 ··· 2 1
when a < b. Unfortunately the same cannot be said when a > b. In particular, consider the pair
of (7,5) noncrossing partitions (P,Q) where P = {{1,4},{2,3}}, each with rank 3, and Q has a
single nontrivial block {4} with rank 1. The Dyck path NNNENNNEEENE gives rise to this
partition. However, the reflected partition is not in NC(a,b) since 1 would be both the maximal
element of a nontrivial block of Q and preced the minimal element of a block of P. By Proposition
3.1, this is impossible.
One might wonder whether a modified reflection for the Q blocks would yield better results. In
particular, what if we use the usual reflection operator on P, but use one of the following shifted
reflection operators on Q:
(cid:18) (cid:19)
1 2 ··· b−2 b−1
rfn(cid:48) =
1 b−1 ··· 3 2
(cid:18) (cid:19)
1 2 ··· n−1 n
rfn(cid:48)(cid:48) =
b−2 b−3 ··· 1 b−1
Neither of the approaches works. To rule out the first, consider the 11,7 noncrossing par-
tition (P,Q) where P = {{1,5,6},{2,4},{3}} with ranks 3, 2, and 3 respectively, and Q =
RATIONAL NONCROSSING PARTITIONS FOR ALL COPRIME PAIRS 9
{{1,4},{5},{6}} each with rank 1. The pair (rfn(P),rfn(cid:48)(Q)) is not in NC(11,7). To rule out
the second, consider the 11,6 noncrossing partition (P,Q) where P = {{1,2,3,5},{4}} with ranks
8 and 2 respectively, and Q = {{4}} with rank 1. The pair (rfn(P),rfn(cid:48)(cid:48)(Q)) is not in NC(11,7).
4. Rank Sequences of Rational Noncrossing Partitions
Let D be a Dyck path such that π(D) is the labeled pair of noncrossing partitions (P,Q). If
B is a block of P, we define rankD(B) to be the length of the vertical run preceding min(B) in
P
D. If B is a block of Q, we define rankD(B) to be the length of the vertical run above max(B)
Q
in D. However, the underlying Dyck path D is almost always clear from context. When this is
the case, we shall simply write rank (B) and rank (B). Given an a,b-Dyck path D such that
P Q
π(D) = (P,Q) ∈ NC(a,b), we define the associated P and Q rank sequences, denoted S and S
P Q
as follows:
(cid:40)
rank (B) if i = min(B) for some B ∈ P
P
S := (p ,p ,...,p ) where p =
P 1 2 b−1 i
0 otherwise
(cid:40)
rank (B) if i = max(B) for some B ∈ Q
Q
S := (q ,q ,...,q ) where q =
Q 1 2 b−1 i
0 otherwise.
In other words, given (P,Q) ∈ NC(a,b) we have π−1(P,Q) = D where
D = Np1ENmax(p2,q1)E···Nmax(pb−1,qb−2)ENqb−1E.
More generally, we will simply define the rank sequence of (P,Q) to be the sequence given by
R(P,Q) := (p ,max(p ,q ),··· ,max(p ,q ),q ).
1 2 1 b−1 b−2 b−1
For example, looking back at the partitions shown in Figure 2 we have S = (3,0,2,3,0,0),
P
S = (0,0,0,0,1,1), and R(P,Q) = (3,0,2,3,0,1,1).
Q
Proposition 4.1. Let a and b be coprime, D be an a,b-Dyck path and π(D) = (P,Q) ∈ NC(a,b).
If B is a block of P, then
rankD(B) = rankrot(D)(rot(B)).
P rot(P)
If B is a block of Q, then
rankD(B) = rankrot(D)(rot(B)).
Q rot(Q)
Proof. It will suffice to consider instead the inverse rotation operator rot(cid:48) defined for a,b-Dyck
paths. This operator preserves vertical run lengths and the underlying block structure of both P
and Q. Preservation of rank is clear unless B contains 1, since rot(cid:48) just subtracts 1 from every
index modulo b − 1. If B is in P and contains 1, then by definition of rot(cid:48), we translate the
entire initial vertical run sequence so it immediately precedes the next element in B, after rot(cid:48)
is applied, so the rank is preserved. If B is in Q and contains 1, then the Q-rise preceding the
maximal element in B is translated to the vertical run preceding the terminal east step in the path.
Thus, the rankD(B) = rankrot(D)(B(cid:48)) where B(cid:48) is the block in rot(Q) coming from rot(D) which
Q rot(Q)
contains b−1. By Proposition 3.6, we have B = B(cid:48), so the rank is again preserved. (cid:3)
Now we show how block ranks respect cardinality under the operation of merging blocks. This is
a generalization of Proposition 3.9 in [5]. In the case where a < b, there are no nontrivial Q blocks
and merging P blocks of (P,Q) ∈ NC(a,b) always yields another a,b-noncrossing partition. When
a > b the result is similar. If there are no nontrivial Q blocks then merging blocks of P yields
another a,b-noncrossing partition. When there are nontrivial blocks of Q, they must sometimes be
split to allow for the merge. This is made precise in the following proposition. An example follows
the end of the proof, which will help clarify the merging operation.
10 MICHELLEBODNAR
Lemma 4.2. Let a and b be coprime positive integers and D be an a,b-Dyck path such that π(D) =
(P,Q) ∈ NC(a,b), and B and B(cid:48) be two blocks of P such that min(B) < min(B(cid:48)). Let P(cid:48) be the
result of replacing B and B(cid:48) in P by B ∪ B(cid:48). If P(cid:48) is a noncrossing partition, then (P(cid:48),Q(cid:48)) ∈
NC(a,b) where Q(cid:48) is obtained as follows: if min(B) ≤ p ≤ min(B(cid:48))−1 and p is in a nontrivial
block B(cid:48)(cid:48) of Q such that max(B(cid:48)(cid:48)) ≥ max(B(cid:48)), remove p from B(cid:48)(cid:48) and put it in a trivial Q block of
rank 0. Further, if D(cid:48) denotes the Dyck path such that π(D) = (P(cid:48),Q(cid:48)), we also have that
rankD(cid:48)(B∪B(cid:48)) = rankD(B)+rankD(B(cid:48))
P(cid:48) P P
and all other ranks are preserved.
Proof. Let D denote the Dyck path such that π(D) = (P,Q). The Dyck path operation which
merges B and B(cid:48) consists of removing the vertical run of length rank(B(cid:48)) atop min(B(cid:48))−1, and
adding rank(B(cid:48)) north steps to the vertical run atop min(B) − 1. We will now verify that this
indeed gives the desired result. Let D(cid:48) denote the Dyck path which results from applying this
operation to D. The only lasers (cid:96)(p) which are potentially affected by this operation are those
such that min(B) − 1 ≤ p ≤ min(B(cid:48)) − 1. For now, assume p (cid:54)= min(B) − 1. If (cid:96)(p) hits west
of min(B(cid:48))−1 in D then it is unchanged in D(cid:48), so we need only consider the case where it hits
east of min(B(cid:48))−1. Observe that the horizontal distance from (cid:96)(min(B(cid:48))−1) and (cid:96)(P) is at most
1. To see this, suppose it were greater than 1. Then there would exist a label q > max(B(cid:48)) such
that (cid:96)(min(B(cid:48))−1) hits D west of q and (cid:96)(p) hits D east of q. Let B(cid:48)(cid:48) be the block containing
q. Then min(B(cid:48)) ≤ min(B(cid:48)(cid:48)) < max(B(cid:48)) < q ≤ max(B(cid:48)(cid:48)) which contradicts the fact that P is
noncrossing. This implies that in D(cid:48), all such lasers hit the interior of the east step immediately
following max(B(cid:48)), so that the block structure and ranks of other blocks of P remains unchanged.
If it should happen that p is in a block B(cid:48)(cid:48) of Q such that max(B(cid:48)(cid:48)) ≥ max(B(cid:48)) then (cid:96)(p) no longer
hits just east of max(B(cid:48)(cid:48)), so it should be removed from that block.
Finally, consider the case where p = min(B) − 1. If (cid:96)(p) hits D east of (cid:96)(min(B(cid:48)) − 1) then
(cid:96)(p) is the same laser in D and D(cid:48). Since (cid:96)(min(B(cid:48)) − 1) disappears, all labels of B(cid:48) become
visible to labels of B, so the blocks union and the ranks sum, as desired. Now suppose (cid:96)(p) hits
D west of (cid:96)(min(B(cid:48))−1). Let C denote the block containing min(B(cid:48))−1. Then we must have
min(C) ≤ min(B)−1 < min(B(cid:48))−1, which implies that merging B and B(cid:48) would create a crossing,
a contradiction. (cid:3)
For example, consider once again the 10,7-Dyck path from Figure 2, along with its associated
noncrossing partitions P and Q. Suppose we would like to merge the blocks B = {1,2} and
B(cid:48) = {3,6} in P. Doing so gives the partition P(cid:48) = {{1,2,3,6},{4,5}} which is indeed non-
crossing, so (P(cid:48),Q(cid:48)) ∈ NC(a,b). To find Q(cid:48), we must look for elements of nontrivial Q blocks
between min(B) = 1 and min(B(cid:48))−1 = 2. There is one such element, 2, contained in the Q block
B(cid:48)(cid:48) = {2,6}. Furthermore, we have max(B(cid:48)(cid:48)) = 6 ≥ max(B(cid:48)). According to the proposition, we re-
move 2 from {2,6} and put it in a trivial Q block of rank 0. Thus, Q(cid:48) = {{1},{2},{3,5},{4},{6}}.
The nontrivial blocks of Q(cid:48) and {3,5} and {6}, each of rank 1.
Next let’s examine how we need to modify D to obtain D(cid:48), where π(D(cid:48)) = (P(cid:48),Q(cid:48)). We remove
the vertical run of length 2 = rank({3,6}) above 2 and adding north steps to the vertical run atop
0 = min({1,2})−1. The new path D(cid:48), along with P(cid:48) and Q(cid:48), each with rank labels shown, are
shown below in Figure 6.
We now discuss the problem of determining whether an arbitrary labeled pair of noncrossing
partitions is in fact a member of NC(a,b). This will be a generalization of the characterization
provided by Proposition 3.5 of [5]. First, we will define a partial order (cid:22) on the blocks of any pair
(P,Q) of noncrossing partitions by